Probing the entanglement of operator growth

TL;DR

The paper addresses operator growth in quantum systems with Lie symmetry (SU(1,1) and SU(2)) by developing a symmetry-based framework built on two-mode coherent states to compute Krylov complexity and entanglement diagnostics. It shows that one can extract the relevant Krylov coefficients and quantum-information quantities directly from symmetry, effectively bypassing the Lanczos algorithm. For SU(1,1), Krylov complexity grows exponentially in time with late-time linear growth in the hopping coefficients, while SU(2) yields finite Krylov growth due to a finite Hilbert space; entanglement measures exhibit universal late-time trends with additional representation-dependent features. The approach provides a versatile toolkit that could extend to other symmetry groups and offers potential links to holography and the geometric interpretation of coherent-state manifolds in the context of operator growth.

Abstract

In this work we probe the operator growth for systems with Lie symmetry using tools from quantum information. Namely, we investigate the Krylov complexity, entanglement negativity, von Neumann entropy and capacity of entanglement for systems with SU(1,1) and SU(2) symmetry. Our main tools are two-mode coherent states, whose properties allow us to study the operator growth and its entanglement structure for any system in a discrete series representation of the groups under consideration. Our results verify that the quantities of interest exhibit certain universal features in agreement with the universal operator growth hypothesis. Moreover, we illustrate the utility of this approach relying on symmetry as it significantly facilitates the calculation of quantities probing operator growth. In particular, we argue that the use of the Lanczos algorithm, which has been the most important tool in the study of operator growth so far, can be circumvented and all the essential information can be extracted directly from symmetry arguments.

Figures (7)

• Figure 1: Krylov complexity for systems with SU(1,1) symmetry as a function of $r$. The curves depicted are for $n_0=0,1,2,3,4,5$. We observe that for bigger $n_0$ the exponential rise of the complexity starts sooner.
• Figure 2: $\varphi_n$ as function of $n$. The chosen parameters for this graph are $n_0=2$ and $r$ takes values from 1 to 2.6 in steps of 0.2. This is essentially the spreading of the wavefunction that was observed in earlier works.
• Figure 3: Left: Comparison of the logarithmic negativity as a function of $r$ for $n_0=1$ between the exact solution (continuous curve) and the approximated solution (dashed curve). We maintain this way of representing exact results using continuous curves and approximate using dashed throughout the rest of this work. Right: Logarithmic negativity as a function of $r$ for $n_0=1,2,7,12,17,22$. Clearly, the features of the approximated functions cannot be trusted for early times as they become negative. However, for late times when our approximation becomes relevant, we observe a linear trend both for the exact and approximated solutions.
• Figure 4: Left: Von Neumann entropy as function of the parameter $r$ for $n_0=0,1,2,3,4$. Once again we observe a universal linear trend for late times, consistent with our expectations. Right: Capacity of entanglement as a function of the parameter $r$ for $n_0=0,1,2,3,4$. The late time behavior is again universal but its quantitative precision is unclear. Also notice that for early times we observe the development of local minima and maxima, which we comment on further in the main text.
• Figure 5: Krylov complexity as a function of $\chi$ for $\frac{1}{2}\leq j\leq 5$. We observe that in all case the Krylov Complexity grows to its maximum value at $\chi=\frac{\pi}{2}$ as expected.